Propagators from integral representations of Green`s functions

Question

I'm working on an article about propagators from int. representations of Green`s functions for several N-dimensional potential(all this is done in an N-dimensional Euclidian space). Potentials like the free-particle, harmonic oscillator, Coulomb and Poschl-Teller. I started from the radial equation which is satisfied by the Qth partial-wave Green's function

$$\biggl[E+\frac{1}{2r^{N-1}}\frac{\partial}{\partial r}r^{N-1}\frac{\partial}{\partial r}-\frac{Q(Q+N-2)}{2r^{2}}-V(r)\biggr]G_{Q}^{N}[r,r',E]=\frac{\delta(r-r')}{(rr')^{N/2-1/2}}$$

As the usual procedure goes, the construction of the Green's function is

$$G[r,r',E]=\frac{u(r_{-})v(r_{+})}{\frac{1}{2}r^{N-1}W[u,v]}$$

obviously, here the u and v are solutions of the homogeneous equation with the appropiate boundary conditions and W is the Wronskian. With the first three potential I had no problem to arive at the N-dimensional form of the propagator.

But I don't know what method to use for the Poschl-Teller potential, I can't even find the solutions for the homogeneous equation with this given potential. Do you have any pointers on this problem?I tried the classical method used to solve the Schrodingers eq. with this potential but I got nowhere with it.

Thanks.

Answer 1

The Pöschl–Teller potential is one of the few exactly solvable models out there, but it's not one with particularly nice solutions. I remember looking at it briefly a long while back in relation to coherent states and supersymmetric quantum mechanics. Its energy eigenstates are a class of 2F1 hypergeometric functions (or generalized Legendre functions in a special case).

I don't know enough to directly help you, but maybe some references will be of use. If not, you can clarify what you want by editing your question.

So, you can find solutions to the homogeneous differential equation and some references in (for example) Supersymmetric partners of the trigonometric Poschl-Teller potentials.

As for calculating the Green's function, I found two papers that use different methods and that might be useful:
Summing the spectral representations of Poschl–Teller and Rosen–Morse fixed-energy amplitudes;
Classification of Solvable Feynman Path Integrals (this is just a talk, for more details, follow the references).